Let $u_t+f(u)_x=0$ be a strictly hyperbolic, genuinely nonlinear system of conservation laws of Temple class. In this paper, a continuous semigroup of solutions is constructed on a domain of $L^\infty$ functions, with possibly unbounded variation. Trajectories depend Lipschitz continuously on the initial data, in the $L^1$ distance. Moreover, we show that a weak solution of the Cauchy problem coincides with the corresponding semigroup trajectory if and only if it satisfies an entropy condition of Oleinik type, concerning the decay of positive waves.

}, url = {http://hdl.handle.net/1963/3256}, author = {Alberto Bressan and Paola Goatin} } @article {1999, title = {Oleinik type estimates and uniqueness for n x n conservation laws}, journal = {J. Differential Equations 156 (1999), no. 1, 26--49}, number = {SISSA;150/97/M}, year = {1999}, publisher = {Elsevier}, abstract = {Let $u_t+f(u)_x=0$ be a strictly hyperbolic $n\\\\times n$ system of conservation laws in one space dimension. Relying on the existence of a semigroup of solutions, we first establish the uniqueness of entropy admissible weak solutions to the Cauchy problem, under a mild assumption on the local oscillation of $u$ in a forward neighborhood of each point in the $t\\\\text{-}x$ plane. In turn, this yields the uniqueness of weak solutions which satisfy a decay estimate on positive waves of genuinely nonlinear families, thus extending a classical result proved by Oleinik in the scalar case.}, doi = {10.1006/jdeq.1998.3606}, url = {http://hdl.handle.net/1963/3375}, author = {Alberto Bressan and Paola Goatin} }